Main:z from

Percentage Accurate: 91.7% → 99.1%

Time: 25.7s

Alternatives: 21

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 91.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{z + 1}\\ t_4 := t\_3 - \sqrt{z}\\ t_5 := \sqrt{1 + y}\\ t_6 := t\_4 + \left(\left(t\_5 - \sqrt{y}\right) + \left(t\_2 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_6 \leq 1.0002:\\ \;\;\;\;\left(t\_4 + \mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{x} + t\_2}\right)\right) + \left(t\_1 - \sqrt{t}\right)\\ \mathbf{elif}\;t\_6 \leq 2.99999995:\\ \;\;\;\;\left(\left(\frac{1}{t\_3 + \sqrt{z}} + t\_5\right) + t\_2\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{1}{\sqrt{t} + t\_1} + t\_3\right) + t\_5\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ t 1.0)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (- t_3 (sqrt z)))
        (t_5 (sqrt (+ 1.0 y)))
        (t_6 (+ t_4 (+ (- t_5 (sqrt y)) (- t_2 (sqrt x))))))
   (if (<= t_6 1.0002)
     (+
      (+ t_4 (fma (sqrt (/ 1.0 y)) 0.5 (/ 1.0 (+ (sqrt x) t_2))))
      (- t_1 (sqrt t)))
     (if (<= t_6 2.99999995)
       (- (+ (+ (/ 1.0 (+ t_3 (sqrt z))) t_5) t_2) (+ (sqrt y) (sqrt x)))
       (+
        (-
         (+ (+ (/ 1.0 (+ (sqrt t) t_1)) t_3) t_5)
         (+ (+ (sqrt z) (sqrt y)) (sqrt x)))
        1.0)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0));
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((z + 1.0));
	double t_4 = t_3 - sqrt(z);
	double t_5 = sqrt((1.0 + y));
	double t_6 = t_4 + ((t_5 - sqrt(y)) + (t_2 - sqrt(x)));
	double tmp;
	if (t_6 <= 1.0002) {
		tmp = (t_4 + fma(sqrt((1.0 / y)), 0.5, (1.0 / (sqrt(x) + t_2)))) + (t_1 - sqrt(t));
	} else if (t_6 <= 2.99999995) {
		tmp = (((1.0 / (t_3 + sqrt(z))) + t_5) + t_2) - (sqrt(y) + sqrt(x));
	} else {
		tmp = ((((1.0 / (sqrt(t) + t_1)) + t_3) + t_5) - ((sqrt(z) + sqrt(y)) + sqrt(x))) + 1.0;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(t + 1.0))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(t_3 - sqrt(z))
	t_5 = sqrt(Float64(1.0 + y))
	t_6 = Float64(t_4 + Float64(Float64(t_5 - sqrt(y)) + Float64(t_2 - sqrt(x))))
	tmp = 0.0
	if (t_6 <= 1.0002)
		tmp = Float64(Float64(t_4 + fma(sqrt(Float64(1.0 / y)), 0.5, Float64(1.0 / Float64(sqrt(x) + t_2)))) + Float64(t_1 - sqrt(t)));
	elseif (t_6 <= 2.99999995)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(t_3 + sqrt(z))) + t_5) + t_2) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(t) + t_1)) + t_3) + t_5) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x))) + 1.0);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 + N[(N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 1.0002], N[(N[(t$95$4 + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.99999995], N[(N[(N[(N[(1.0 / N[(t$95$3 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$5), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.0002

   1. Initial program 85.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. flip-- N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 88.1%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(y + 1\right) - y, \sqrt{x} + \sqrt{1 + x}, \left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\left(1 + x\right) - x\right)\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\color{blue}{1 + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{1 + y}\right)\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right)} + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\color{blue}{\sqrt{y}} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \color{blue}{\sqrt{1 + y}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{y + 1}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 70.1

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{y + 1}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 70.1%

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{y + 1}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{y}}}, \frac{1}{2}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{y}}}, \frac{1}{2}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\color{blue}{\sqrt{1 + x}} + \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 68.0

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{x + 1} + \color{blue}{\sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied rewrites 68.0%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.0002 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.99999994999999986

   1. Initial program 97.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 98.3

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 98.3%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   7. Applied rewrites 25.6%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)} \]

   if 2.99999994999999986 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 97.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right)} \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\sqrt{t + 1}} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\sqrt{t + 1} \cdot \color{blue}{\sqrt{t + 1}} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right)} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{\sqrt{t}} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \sqrt{t} \cdot \color{blue}{\sqrt{t}}}{\sqrt{t + 1} + \sqrt{t}} \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right) - t}}{\sqrt{t + 1} + \sqrt{t}} \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

      12. lower-+.f64 98.4

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

   4. Applied rewrites 98.4%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\frac{1}{\sqrt{t + 1} + \sqrt{t}} + \sqrt{z + 1}\right) + \sqrt{y + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.0002:\\ \;\;\;\;\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2.99999995:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{1}{\sqrt{t} + \sqrt{t + 1}} + \sqrt{z + 1}\right) + \sqrt{1 + y}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{t + 1}\\ t_4 := \sqrt{z + 1}\\ t_5 := t\_4 - \sqrt{z}\\ t_6 := t\_3 - \sqrt{t}\\ t_7 := \left(t\_5 + \left(\left(t\_1 - \sqrt{y}\right) + \left(t\_2 - \sqrt{x}\right)\right)\right) + t\_6\\ \mathbf{if}\;t\_7 \leq 1:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + t\_2} + t\_5\right) + t\_6\\ \mathbf{elif}\;t\_7 \leq 3:\\ \;\;\;\;\left(\left(\frac{1}{t\_4 + \sqrt{z}} + t\_1\right) + t\_2\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t\_4 + t\_1\right) + t\_3\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right) + 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (sqrt (+ t 1.0)))
        (t_4 (sqrt (+ z 1.0)))
        (t_5 (- t_4 (sqrt z)))
        (t_6 (- t_3 (sqrt t)))
        (t_7 (+ (+ t_5 (+ (- t_1 (sqrt y)) (- t_2 (sqrt x)))) t_6)))
   (if (<= t_7 1.0)
     (+ (+ (/ 1.0 (+ (sqrt x) t_2)) t_5) t_6)
     (if (<= t_7 3.0)
       (- (+ (+ (/ 1.0 (+ t_4 (sqrt z))) t_1) t_2) (+ (sqrt y) (sqrt x)))
       (+
        (- (+ (+ t_4 t_1) t_3) (+ (+ (+ (sqrt z) (sqrt y)) (sqrt x)) (sqrt t)))
        1.0)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((t + 1.0));
	double t_4 = sqrt((z + 1.0));
	double t_5 = t_4 - sqrt(z);
	double t_6 = t_3 - sqrt(t);
	double t_7 = (t_5 + ((t_1 - sqrt(y)) + (t_2 - sqrt(x)))) + t_6;
	double tmp;
	if (t_7 <= 1.0) {
		tmp = ((1.0 / (sqrt(x) + t_2)) + t_5) + t_6;
	} else if (t_7 <= 3.0) {
		tmp = (((1.0 / (t_4 + sqrt(z))) + t_1) + t_2) - (sqrt(y) + sqrt(x));
	} else {
		tmp = (((t_4 + t_1) + t_3) - (((sqrt(z) + sqrt(y)) + sqrt(x)) + sqrt(t))) + 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((1.0d0 + x))
    t_3 = sqrt((t + 1.0d0))
    t_4 = sqrt((z + 1.0d0))
    t_5 = t_4 - sqrt(z)
    t_6 = t_3 - sqrt(t)
    t_7 = (t_5 + ((t_1 - sqrt(y)) + (t_2 - sqrt(x)))) + t_6
    if (t_7 <= 1.0d0) then
        tmp = ((1.0d0 / (sqrt(x) + t_2)) + t_5) + t_6
    else if (t_7 <= 3.0d0) then
        tmp = (((1.0d0 / (t_4 + sqrt(z))) + t_1) + t_2) - (sqrt(y) + sqrt(x))
    else
        tmp = (((t_4 + t_1) + t_3) - (((sqrt(z) + sqrt(y)) + sqrt(x)) + sqrt(t))) + 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((1.0 + x));
	double t_3 = Math.sqrt((t + 1.0));
	double t_4 = Math.sqrt((z + 1.0));
	double t_5 = t_4 - Math.sqrt(z);
	double t_6 = t_3 - Math.sqrt(t);
	double t_7 = (t_5 + ((t_1 - Math.sqrt(y)) + (t_2 - Math.sqrt(x)))) + t_6;
	double tmp;
	if (t_7 <= 1.0) {
		tmp = ((1.0 / (Math.sqrt(x) + t_2)) + t_5) + t_6;
	} else if (t_7 <= 3.0) {
		tmp = (((1.0 / (t_4 + Math.sqrt(z))) + t_1) + t_2) - (Math.sqrt(y) + Math.sqrt(x));
	} else {
		tmp = (((t_4 + t_1) + t_3) - (((Math.sqrt(z) + Math.sqrt(y)) + Math.sqrt(x)) + Math.sqrt(t))) + 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt((1.0 + x))
	t_3 = math.sqrt((t + 1.0))
	t_4 = math.sqrt((z + 1.0))
	t_5 = t_4 - math.sqrt(z)
	t_6 = t_3 - math.sqrt(t)
	t_7 = (t_5 + ((t_1 - math.sqrt(y)) + (t_2 - math.sqrt(x)))) + t_6
	tmp = 0
	if t_7 <= 1.0:
		tmp = ((1.0 / (math.sqrt(x) + t_2)) + t_5) + t_6
	elif t_7 <= 3.0:
		tmp = (((1.0 / (t_4 + math.sqrt(z))) + t_1) + t_2) - (math.sqrt(y) + math.sqrt(x))
	else:
		tmp = (((t_4 + t_1) + t_3) - (((math.sqrt(z) + math.sqrt(y)) + math.sqrt(x)) + math.sqrt(t))) + 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(t + 1.0))
	t_4 = sqrt(Float64(z + 1.0))
	t_5 = Float64(t_4 - sqrt(z))
	t_6 = Float64(t_3 - sqrt(t))
	t_7 = Float64(Float64(t_5 + Float64(Float64(t_1 - sqrt(y)) + Float64(t_2 - sqrt(x)))) + t_6)
	tmp = 0.0
	if (t_7 <= 1.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_2)) + t_5) + t_6);
	elseif (t_7 <= 3.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(t_4 + sqrt(z))) + t_1) + t_2) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(Float64(t_4 + t_1) + t_3) - Float64(Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x)) + sqrt(t))) + 1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((1.0 + x));
	t_3 = sqrt((t + 1.0));
	t_4 = sqrt((z + 1.0));
	t_5 = t_4 - sqrt(z);
	t_6 = t_3 - sqrt(t);
	t_7 = (t_5 + ((t_1 - sqrt(y)) + (t_2 - sqrt(x)))) + t_6;
	tmp = 0.0;
	if (t_7 <= 1.0)
		tmp = ((1.0 / (sqrt(x) + t_2)) + t_5) + t_6;
	elseif (t_7 <= 3.0)
		tmp = (((1.0 / (t_4 + sqrt(z))) + t_1) + t_2) - (sqrt(y) + sqrt(x));
	else
		tmp = (((t_4 + t_1) + t_3) - (((sqrt(z) + sqrt(y)) + sqrt(x)) + sqrt(t))) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$5 + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision]}, If[LessEqual[t$95$7, 1.0], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$6), $MachinePrecision], If[LessEqual[t$95$7, 3.0], N[(N[(N[(N[(1.0 / N[(t$95$4 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$4 + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] - N[(N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1

   1. Initial program 77.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. flip-- N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 79.6%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(y + 1\right) - y, \sqrt{x} + \sqrt{1 + x}, \left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\left(1 + x\right) - x\right)\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\color{blue}{1 + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{1 + y}\right)\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right)} + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\color{blue}{\sqrt{y}} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \color{blue}{\sqrt{1 + y}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{y + 1}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 66.9

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{y + 1}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 66.9%

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{y + 1}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 63.8

         \[\leadsto \left(\frac{1}{\sqrt{x + 1} + \color{blue}{\sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied rewrites 63.8%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3

   1. Initial program 96.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 97.4

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 97.4%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   7. Applied rewrites 23.3%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)} \]

   if 3 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 99.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 21.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   8. Applied rewrites 96.8%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{t + 1} + \left(\sqrt{z + 1} + \sqrt{y + 1}\right)\right) - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 1:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 3:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt{z + 1} + \sqrt{1 + y}\right) + \sqrt{t + 1}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{z + 1}\\ t_4 := t\_3 - \sqrt{z}\\ t_5 := \sqrt{t + 1} - \sqrt{t}\\ t_6 := \left(t\_4 + \left(\left(t\_1 - \sqrt{y}\right) + \left(t\_2 - \sqrt{x}\right)\right)\right) + t\_5\\ \mathbf{if}\;t\_6 \leq 1:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + t\_2} + t\_4\right) + t\_5\\ \mathbf{elif}\;t\_6 \leq 2.5:\\ \;\;\;\;\left(\left(\frac{1}{t\_3 + \sqrt{z}} + t\_1\right) + t\_2\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right) + t\_4\right) + t\_5\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (- t_3 (sqrt z)))
        (t_5 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_6 (+ (+ t_4 (+ (- t_1 (sqrt y)) (- t_2 (sqrt x)))) t_5)))
   (if (<= t_6 1.0)
     (+ (+ (/ 1.0 (+ (sqrt x) t_2)) t_4) t_5)
     (if (<= t_6 2.5)
       (- (+ (+ (/ 1.0 (+ t_3 (sqrt z))) t_1) t_2) (+ (sqrt y) (sqrt x)))
       (+ (+ (+ (- 1.0 (sqrt y)) (- 1.0 (sqrt x))) t_4) t_5)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((z + 1.0));
	double t_4 = t_3 - sqrt(z);
	double t_5 = sqrt((t + 1.0)) - sqrt(t);
	double t_6 = (t_4 + ((t_1 - sqrt(y)) + (t_2 - sqrt(x)))) + t_5;
	double tmp;
	if (t_6 <= 1.0) {
		tmp = ((1.0 / (sqrt(x) + t_2)) + t_4) + t_5;
	} else if (t_6 <= 2.5) {
		tmp = (((1.0 / (t_3 + sqrt(z))) + t_1) + t_2) - (sqrt(y) + sqrt(x));
	} else {
		tmp = (((1.0 - sqrt(y)) + (1.0 - sqrt(x))) + t_4) + t_5;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y))
    t_2 = sqrt((1.0d0 + x))
    t_3 = sqrt((z + 1.0d0))
    t_4 = t_3 - sqrt(z)
    t_5 = sqrt((t + 1.0d0)) - sqrt(t)
    t_6 = (t_4 + ((t_1 - sqrt(y)) + (t_2 - sqrt(x)))) + t_5
    if (t_6 <= 1.0d0) then
        tmp = ((1.0d0 / (sqrt(x) + t_2)) + t_4) + t_5
    else if (t_6 <= 2.5d0) then
        tmp = (((1.0d0 / (t_3 + sqrt(z))) + t_1) + t_2) - (sqrt(y) + sqrt(x))
    else
        tmp = (((1.0d0 - sqrt(y)) + (1.0d0 - sqrt(x))) + t_4) + t_5
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + y));
	double t_2 = Math.sqrt((1.0 + x));
	double t_3 = Math.sqrt((z + 1.0));
	double t_4 = t_3 - Math.sqrt(z);
	double t_5 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_6 = (t_4 + ((t_1 - Math.sqrt(y)) + (t_2 - Math.sqrt(x)))) + t_5;
	double tmp;
	if (t_6 <= 1.0) {
		tmp = ((1.0 / (Math.sqrt(x) + t_2)) + t_4) + t_5;
	} else if (t_6 <= 2.5) {
		tmp = (((1.0 / (t_3 + Math.sqrt(z))) + t_1) + t_2) - (Math.sqrt(y) + Math.sqrt(x));
	} else {
		tmp = (((1.0 - Math.sqrt(y)) + (1.0 - Math.sqrt(x))) + t_4) + t_5;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	t_2 = math.sqrt((1.0 + x))
	t_3 = math.sqrt((z + 1.0))
	t_4 = t_3 - math.sqrt(z)
	t_5 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_6 = (t_4 + ((t_1 - math.sqrt(y)) + (t_2 - math.sqrt(x)))) + t_5
	tmp = 0
	if t_6 <= 1.0:
		tmp = ((1.0 / (math.sqrt(x) + t_2)) + t_4) + t_5
	elif t_6 <= 2.5:
		tmp = (((1.0 / (t_3 + math.sqrt(z))) + t_1) + t_2) - (math.sqrt(y) + math.sqrt(x))
	else:
		tmp = (((1.0 - math.sqrt(y)) + (1.0 - math.sqrt(x))) + t_4) + t_5
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(t_3 - sqrt(z))
	t_5 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_6 = Float64(Float64(t_4 + Float64(Float64(t_1 - sqrt(y)) + Float64(t_2 - sqrt(x)))) + t_5)
	tmp = 0.0
	if (t_6 <= 1.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_2)) + t_4) + t_5);
	elseif (t_6 <= 2.5)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(t_3 + sqrt(z))) + t_1) + t_2) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(y)) + Float64(1.0 - sqrt(x))) + t_4) + t_5);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + y));
	t_2 = sqrt((1.0 + x));
	t_3 = sqrt((z + 1.0));
	t_4 = t_3 - sqrt(z);
	t_5 = sqrt((t + 1.0)) - sqrt(t);
	t_6 = (t_4 + ((t_1 - sqrt(y)) + (t_2 - sqrt(x)))) + t_5;
	tmp = 0.0;
	if (t_6 <= 1.0)
		tmp = ((1.0 / (sqrt(x) + t_2)) + t_4) + t_5;
	elseif (t_6 <= 2.5)
		tmp = (((1.0 / (t_3 + sqrt(z))) + t_1) + t_2) - (sqrt(y) + sqrt(x));
	else
		tmp = (((1.0 - sqrt(y)) + (1.0 - sqrt(x))) + t_4) + t_5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$4 + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]}, If[LessEqual[t$95$6, 1.0], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$5), $MachinePrecision], If[LessEqual[t$95$6, 2.5], N[(N[(N[(N[(1.0 / N[(t$95$3 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$5), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1

   1. Initial program 77.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. flip-- N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 79.6%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(y + 1\right) - y, \sqrt{x} + \sqrt{1 + x}, \left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\left(1 + x\right) - x\right)\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\color{blue}{1 + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{1 + y}\right)\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right)} + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\color{blue}{\sqrt{y}} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \color{blue}{\sqrt{1 + y}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{y + 1}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 66.9

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{y + 1}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 66.9%

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{y + 1}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 63.8

         \[\leadsto \left(\frac{1}{\sqrt{x + 1} + \color{blue}{\sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied rewrites 63.8%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.5

   1. Initial program 95.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 96.4

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 96.4%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   7. Applied rewrites 19.3%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)} \]

   if 2.5 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 99.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-sqrt.f64 79.0

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 79.0%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right) + \color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right) + \color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right) + \color{blue}{\sqrt{\frac{1}{y}}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 20.7

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right) + \sqrt{\color{blue}{\frac{1}{y}}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 20.7%

      \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right) + \color{blue}{\sqrt{\frac{1}{y}} \cdot 0.5}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \sqrt{\frac{1}{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \sqrt{\frac{1}{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-sqrt.f64 20.6

          \[\leadsto \left(\left(\left(1 - \color{blue}{\sqrt{x}}\right) + \sqrt{\frac{1}{y}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Applied rewrites 20.6%

       \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \sqrt{\frac{1}{y}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Taylor expanded in y around 0

       \[\leadsto \left(\left(\left(1 - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   13. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \left(\left(\left(1 - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-sqrt.f64 61.0

          \[\leadsto \left(\left(\left(1 - \sqrt{x}\right) + \left(1 - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   14. Applied rewrites 61.0%

       \[\leadsto \left(\left(\left(1 - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 1:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 2.5:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ t_2 := \sqrt{z + 1}\\ t_3 := t\_2 - \sqrt{z}\\ t_4 := t\_3 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_4 \leq 0.0001:\\ \;\;\;\;\left(\left(0.5 \cdot \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{x}} \cdot 0.5\right) + t\_3\right) + t\_1\\ \mathbf{elif}\;t\_4 \leq 1.9998:\\ \;\;\;\;t\_4 + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 + \mathsf{fma}\left(0.5, y, \frac{1}{t\_2 + \sqrt{z}}\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (- t_2 (sqrt z)))
        (t_4
         (+
          t_3
          (+ (- (sqrt (+ 1.0 y)) (sqrt y)) (- (sqrt (+ 1.0 x)) (sqrt x))))))
   (if (<= t_4 0.0001)
     (+ (+ (+ (* 0.5 (sqrt (/ 1.0 y))) (* (sqrt (/ 1.0 x)) 0.5)) t_3) t_1)
     (if (<= t_4 1.9998)
       (+ t_4 t_1)
       (+
        (- (+ 2.0 (fma 0.5 y (/ 1.0 (+ t_2 (sqrt z))))) (+ (sqrt y) (sqrt x)))
        t_1)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((t + 1.0)) - sqrt(t);
	double t_2 = sqrt((z + 1.0));
	double t_3 = t_2 - sqrt(z);
	double t_4 = t_3 + ((sqrt((1.0 + y)) - sqrt(y)) + (sqrt((1.0 + x)) - sqrt(x)));
	double tmp;
	if (t_4 <= 0.0001) {
		tmp = (((0.5 * sqrt((1.0 / y))) + (sqrt((1.0 / x)) * 0.5)) + t_3) + t_1;
	} else if (t_4 <= 1.9998) {
		tmp = t_4 + t_1;
	} else {
		tmp = ((2.0 + fma(0.5, y, (1.0 / (t_2 + sqrt(z))))) - (sqrt(y) + sqrt(x))) + t_1;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = Float64(t_2 - sqrt(z))
	t_4 = Float64(t_3 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + Float64(sqrt(Float64(1.0 + x)) - sqrt(x))))
	tmp = 0.0
	if (t_4 <= 0.0001)
		tmp = Float64(Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(sqrt(Float64(1.0 / x)) * 0.5)) + t_3) + t_1);
	elseif (t_4 <= 1.9998)
		tmp = Float64(t_4 + t_1);
	else
		tmp = Float64(Float64(Float64(2.0 + fma(0.5, y, Float64(1.0 / Float64(t_2 + sqrt(z))))) - Float64(sqrt(y) + sqrt(x))) + t_1);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0001], N[(N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 1.9998], N[(t$95$4 + t$95$1), $MachinePrecision], N[(N[(N[(2.0 + N[(0.5 * y + N[(1.0 / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00000000000000005e-4

   1. Initial program 57.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-sqrt.f64 4.9

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 4.9%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right) + \color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right) + \color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right) + \color{blue}{\sqrt{\frac{1}{y}}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 4.9

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right) + \sqrt{\color{blue}{\frac{1}{y}}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 4.9%

      \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right) + \color{blue}{\sqrt{\frac{1}{y}} \cdot 0.5}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}} + \sqrt{\frac{1}{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}} + \sqrt{\frac{1}{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}}} \cdot \frac{1}{2} + \sqrt{\frac{1}{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-/.f64 65.7

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{x}}} \cdot 0.5 + \sqrt{\frac{1}{y}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Applied rewrites 65.7%

       \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} + \sqrt{\frac{1}{y}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.00000000000000005e-4 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.9998

   1. Initial program 95.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   if 1.9998 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 98.3

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 98.3%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot y + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot y + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} + 1\right) + \mathsf{fma}\left(0.5, y, \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(\left(2 + \left(\frac{1}{2} \cdot y + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 50.1%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.5, y, \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right) + 2\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 0.0001:\\ \;\;\;\;\left(\left(0.5 \cdot \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{x}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.9998:\\ \;\;\;\;\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 + \mathsf{fma}\left(0.5, y, \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := t\_1 - \sqrt{z}\\ t_3 := \sqrt{1 + y}\\ t_4 := \sqrt{1 + x}\\ t_5 := t\_2 + \left(\left(t\_3 - \sqrt{y}\right) + \left(t\_4 - \sqrt{x}\right)\right)\\ t_6 := \sqrt{t + 1}\\ \mathbf{if}\;t\_5 \leq 0.0005:\\ \;\;\;\;\left(\left(0.5 \cdot \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{x}} \cdot 0.5\right) + t\_2\right) + \left(t\_6 - \sqrt{t}\right)\\ \mathbf{elif}\;t\_5 \leq 2.001:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, t\_3\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + t\_4\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{1}{\sqrt{t} + t\_6} + t\_1\right) + t\_3\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0)))
        (t_2 (- t_1 (sqrt z)))
        (t_3 (sqrt (+ 1.0 y)))
        (t_4 (sqrt (+ 1.0 x)))
        (t_5 (+ t_2 (+ (- t_3 (sqrt y)) (- t_4 (sqrt x)))))
        (t_6 (sqrt (+ t 1.0))))
   (if (<= t_5 0.0005)
     (+
      (+ (+ (* 0.5 (sqrt (/ 1.0 y))) (* (sqrt (/ 1.0 x)) 0.5)) t_2)
      (- t_6 (sqrt t)))
     (if (<= t_5 2.001)
       (+ (- (fma (sqrt (/ 1.0 z)) 0.5 t_3) (+ (sqrt y) (sqrt x))) t_4)
       (+
        (-
         (+ (+ (/ 1.0 (+ (sqrt t) t_6)) t_1) t_3)
         (+ (+ (sqrt z) (sqrt y)) (sqrt x)))
        1.0)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = t_1 - sqrt(z);
	double t_3 = sqrt((1.0 + y));
	double t_4 = sqrt((1.0 + x));
	double t_5 = t_2 + ((t_3 - sqrt(y)) + (t_4 - sqrt(x)));
	double t_6 = sqrt((t + 1.0));
	double tmp;
	if (t_5 <= 0.0005) {
		tmp = (((0.5 * sqrt((1.0 / y))) + (sqrt((1.0 / x)) * 0.5)) + t_2) + (t_6 - sqrt(t));
	} else if (t_5 <= 2.001) {
		tmp = (fma(sqrt((1.0 / z)), 0.5, t_3) - (sqrt(y) + sqrt(x))) + t_4;
	} else {
		tmp = ((((1.0 / (sqrt(t) + t_6)) + t_1) + t_3) - ((sqrt(z) + sqrt(y)) + sqrt(x))) + 1.0;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = Float64(t_1 - sqrt(z))
	t_3 = sqrt(Float64(1.0 + y))
	t_4 = sqrt(Float64(1.0 + x))
	t_5 = Float64(t_2 + Float64(Float64(t_3 - sqrt(y)) + Float64(t_4 - sqrt(x))))
	t_6 = sqrt(Float64(t + 1.0))
	tmp = 0.0
	if (t_5 <= 0.0005)
		tmp = Float64(Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / y))) + Float64(sqrt(Float64(1.0 / x)) * 0.5)) + t_2) + Float64(t_6 - sqrt(t)));
	elseif (t_5 <= 2.001)
		tmp = Float64(Float64(fma(sqrt(Float64(1.0 / z)), 0.5, t_3) - Float64(sqrt(y) + sqrt(x))) + t_4);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(t) + t_6)) + t_1) + t_3) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x))) + 1.0);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 + N[(N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, 0.0005], N[(N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(t$95$6 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.001], N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$3), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], N[(N[(N[(N[(N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 5.0000000000000001e-4

   1. Initial program 58.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-sqrt.f64 4.9

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 4.9%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right) + \color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right) + \color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right) + \color{blue}{\sqrt{\frac{1}{y}}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 4.9

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right) + \sqrt{\color{blue}{\frac{1}{y}}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 4.9%

      \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right) + \color{blue}{\sqrt{\frac{1}{y}} \cdot 0.5}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} + \sqrt{\frac{1}{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}} + \sqrt{\frac{1}{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}} + \sqrt{\frac{1}{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}}} \cdot \frac{1}{2} + \sqrt{\frac{1}{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-/.f64 65.2

          \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{x}}} \cdot 0.5 + \sqrt{\frac{1}{y}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Applied rewrites 65.2%

       \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} + \sqrt{\frac{1}{y}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 5.0000000000000001e-4 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00099999999999989

   1. Initial program 96.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 5.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 17.3%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{y + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in z around inf

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 21.8%

       \[\leadsto \sqrt{x + 1} + \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   if 2.00099999999999989 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right)} \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\sqrt{t + 1}} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\sqrt{t + 1} \cdot \color{blue}{\sqrt{t + 1}} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right)} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{\sqrt{t}} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \sqrt{t} \cdot \color{blue}{\sqrt{t}}}{\sqrt{t + 1} + \sqrt{t}} \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right) - t}}{\sqrt{t + 1} + \sqrt{t}} \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

      12. lower-+.f64 98.6

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

   4. Applied rewrites 98.6%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   7. Applied rewrites 98.1%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\frac{1}{\sqrt{t + 1} + \sqrt{t}} + \sqrt{z + 1}\right) + \sqrt{y + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 0.0005:\\ \;\;\;\;\left(\left(0.5 \cdot \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{x}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2.001:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + \sqrt{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{1}{\sqrt{t} + \sqrt{t + 1}} + \sqrt{z + 1}\right) + \sqrt{1 + y}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{z + 1}\\ t_3 := t\_2 - \sqrt{z}\\ t_4 := \sqrt{t + 1}\\ t_5 := \sqrt{1 + y}\\ t_6 := t\_3 + \left(\left(t\_5 - \sqrt{y}\right) + \left(t\_1 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_6 \leq 1:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + t\_1} + t\_3\right) + \left(t\_4 - \sqrt{t}\right)\\ \mathbf{elif}\;t\_6 \leq 2.99999995:\\ \;\;\;\;\left(\left(\frac{1}{t\_2 + \sqrt{z}} + t\_5\right) + t\_1\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{1}{\sqrt{t} + t\_4} + t\_2\right) + t\_5\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (- t_2 (sqrt z)))
        (t_4 (sqrt (+ t 1.0)))
        (t_5 (sqrt (+ 1.0 y)))
        (t_6 (+ t_3 (+ (- t_5 (sqrt y)) (- t_1 (sqrt x))))))
   (if (<= t_6 1.0)
     (+ (+ (/ 1.0 (+ (sqrt x) t_1)) t_3) (- t_4 (sqrt t)))
     (if (<= t_6 2.99999995)
       (- (+ (+ (/ 1.0 (+ t_2 (sqrt z))) t_5) t_1) (+ (sqrt y) (sqrt x)))
       (+
        (-
         (+ (+ (/ 1.0 (+ (sqrt t) t_4)) t_2) t_5)
         (+ (+ (sqrt z) (sqrt y)) (sqrt x)))
        1.0)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((z + 1.0));
	double t_3 = t_2 - sqrt(z);
	double t_4 = sqrt((t + 1.0));
	double t_5 = sqrt((1.0 + y));
	double t_6 = t_3 + ((t_5 - sqrt(y)) + (t_1 - sqrt(x)));
	double tmp;
	if (t_6 <= 1.0) {
		tmp = ((1.0 / (sqrt(x) + t_1)) + t_3) + (t_4 - sqrt(t));
	} else if (t_6 <= 2.99999995) {
		tmp = (((1.0 / (t_2 + sqrt(z))) + t_5) + t_1) - (sqrt(y) + sqrt(x));
	} else {
		tmp = ((((1.0 / (sqrt(t) + t_4)) + t_2) + t_5) - ((sqrt(z) + sqrt(y)) + sqrt(x))) + 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((z + 1.0d0))
    t_3 = t_2 - sqrt(z)
    t_4 = sqrt((t + 1.0d0))
    t_5 = sqrt((1.0d0 + y))
    t_6 = t_3 + ((t_5 - sqrt(y)) + (t_1 - sqrt(x)))
    if (t_6 <= 1.0d0) then
        tmp = ((1.0d0 / (sqrt(x) + t_1)) + t_3) + (t_4 - sqrt(t))
    else if (t_6 <= 2.99999995d0) then
        tmp = (((1.0d0 / (t_2 + sqrt(z))) + t_5) + t_1) - (sqrt(y) + sqrt(x))
    else
        tmp = ((((1.0d0 / (sqrt(t) + t_4)) + t_2) + t_5) - ((sqrt(z) + sqrt(y)) + sqrt(x))) + 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((z + 1.0));
	double t_3 = t_2 - Math.sqrt(z);
	double t_4 = Math.sqrt((t + 1.0));
	double t_5 = Math.sqrt((1.0 + y));
	double t_6 = t_3 + ((t_5 - Math.sqrt(y)) + (t_1 - Math.sqrt(x)));
	double tmp;
	if (t_6 <= 1.0) {
		tmp = ((1.0 / (Math.sqrt(x) + t_1)) + t_3) + (t_4 - Math.sqrt(t));
	} else if (t_6 <= 2.99999995) {
		tmp = (((1.0 / (t_2 + Math.sqrt(z))) + t_5) + t_1) - (Math.sqrt(y) + Math.sqrt(x));
	} else {
		tmp = ((((1.0 / (Math.sqrt(t) + t_4)) + t_2) + t_5) - ((Math.sqrt(z) + Math.sqrt(y)) + Math.sqrt(x))) + 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = math.sqrt((z + 1.0))
	t_3 = t_2 - math.sqrt(z)
	t_4 = math.sqrt((t + 1.0))
	t_5 = math.sqrt((1.0 + y))
	t_6 = t_3 + ((t_5 - math.sqrt(y)) + (t_1 - math.sqrt(x)))
	tmp = 0
	if t_6 <= 1.0:
		tmp = ((1.0 / (math.sqrt(x) + t_1)) + t_3) + (t_4 - math.sqrt(t))
	elif t_6 <= 2.99999995:
		tmp = (((1.0 / (t_2 + math.sqrt(z))) + t_5) + t_1) - (math.sqrt(y) + math.sqrt(x))
	else:
		tmp = ((((1.0 / (math.sqrt(t) + t_4)) + t_2) + t_5) - ((math.sqrt(z) + math.sqrt(y)) + math.sqrt(x))) + 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = Float64(t_2 - sqrt(z))
	t_4 = sqrt(Float64(t + 1.0))
	t_5 = sqrt(Float64(1.0 + y))
	t_6 = Float64(t_3 + Float64(Float64(t_5 - sqrt(y)) + Float64(t_1 - sqrt(x))))
	tmp = 0.0
	if (t_6 <= 1.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_1)) + t_3) + Float64(t_4 - sqrt(t)));
	elseif (t_6 <= 2.99999995)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(t_2 + sqrt(z))) + t_5) + t_1) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(t) + t_4)) + t_2) + t_5) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x))) + 1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = sqrt((z + 1.0));
	t_3 = t_2 - sqrt(z);
	t_4 = sqrt((t + 1.0));
	t_5 = sqrt((1.0 + y));
	t_6 = t_3 + ((t_5 - sqrt(y)) + (t_1 - sqrt(x)));
	tmp = 0.0;
	if (t_6 <= 1.0)
		tmp = ((1.0 / (sqrt(x) + t_1)) + t_3) + (t_4 - sqrt(t));
	elseif (t_6 <= 2.99999995)
		tmp = (((1.0 / (t_2 + sqrt(z))) + t_5) + t_1) - (sqrt(y) + sqrt(x));
	else
		tmp = ((((1.0 / (sqrt(t) + t_4)) + t_2) + t_5) - ((sqrt(z) + sqrt(y)) + sqrt(x))) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 + N[(N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 1.0], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.99999995], N[(N[(N[(N[(1.0 / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$1), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$5), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1

   1. Initial program 86.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. flip-- N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 87.6%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(y + 1\right) - y, \sqrt{x} + \sqrt{1 + x}, \left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\left(1 + x\right) - x\right)\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\color{blue}{1 + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{1 + y}\right)\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right)} + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\color{blue}{\sqrt{y}} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \color{blue}{\sqrt{1 + y}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{y + 1}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 70.1

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{y + 1}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 70.1%

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{y + 1}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 68.1

         \[\leadsto \left(\frac{1}{\sqrt{x + 1} + \color{blue}{\sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied rewrites 68.1%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.99999994999999986

   1. Initial program 96.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 97.2

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 97.2%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   7. Applied rewrites 24.9%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)} \]

   if 2.99999994999999986 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 97.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right)} \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\sqrt{t + 1}} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\sqrt{t + 1} \cdot \color{blue}{\sqrt{t + 1}} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right)} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{\sqrt{t}} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \sqrt{t} \cdot \color{blue}{\sqrt{t}}}{\sqrt{t + 1} + \sqrt{t}} \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right) - t}}{\sqrt{t + 1} + \sqrt{t}} \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

      12. lower-+.f64 98.4

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

   4. Applied rewrites 98.4%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\frac{1}{\sqrt{t + 1} + \sqrt{t}} + \sqrt{z + 1}\right) + \sqrt{y + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2.99999995:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{1}{\sqrt{t} + \sqrt{t + 1}} + \sqrt{z + 1}\right) + \sqrt{1 + y}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 52000000:\\ \;\;\;\;t\_2 + \left(\frac{\left(z + 1\right) - z}{t\_1 + \sqrt{z}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(t\_3 - \sqrt{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 - \sqrt{z}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{x} + t\_3}\right)\right) + t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0)))
        (t_2 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_3 (sqrt (+ 1.0 x))))
   (if (<= y 52000000.0)
     (+
      t_2
      (+
       (/ (- (+ z 1.0) z) (+ t_1 (sqrt z)))
       (+ (- (sqrt (+ 1.0 y)) (sqrt y)) (- t_3 (sqrt x)))))
     (+
      (+ (- t_1 (sqrt z)) (fma (sqrt (/ 1.0 y)) 0.5 (/ 1.0 (+ (sqrt x) t_3))))
      t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = sqrt((t + 1.0)) - sqrt(t);
	double t_3 = sqrt((1.0 + x));
	double tmp;
	if (y <= 52000000.0) {
		tmp = t_2 + ((((z + 1.0) - z) / (t_1 + sqrt(z))) + ((sqrt((1.0 + y)) - sqrt(y)) + (t_3 - sqrt(x))));
	} else {
		tmp = ((t_1 - sqrt(z)) + fma(sqrt((1.0 / y)), 0.5, (1.0 / (sqrt(x) + t_3)))) + t_2;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_3 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (y <= 52000000.0)
		tmp = Float64(t_2 + Float64(Float64(Float64(Float64(z + 1.0) - z) / Float64(t_1 + sqrt(z))) + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + Float64(t_3 - sqrt(x)))));
	else
		tmp = Float64(Float64(Float64(t_1 - sqrt(z)) + fma(sqrt(Float64(1.0 / y)), 0.5, Float64(1.0 / Float64(sqrt(x) + t_3)))) + t_2);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 52000000.0], N[(t$95$2 + N[(N[(N[(N[(z + 1.0), $MachinePrecision] - z), $MachinePrecision] / N[(t$95$1 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.2e7

   1. Initial program 97.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 97.8

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 97.8%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 5.2e7 < y

   1. Initial program 86.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. flip-- N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 89.1%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(y + 1\right) - y, \sqrt{x} + \sqrt{1 + x}, \left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\left(1 + x\right) - x\right)\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\color{blue}{1 + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{1 + y}\right)\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right)} + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\color{blue}{\sqrt{y}} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \color{blue}{\sqrt{1 + y}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{y + 1}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 90.3

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{y + 1}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 90.3%

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{y + 1}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{y}}}, \frac{1}{2}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{y}}}, \frac{1}{2}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\color{blue}{\sqrt{1 + x}} + \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, \frac{1}{2}, \frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 90.8

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{x + 1} + \color{blue}{\sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied rewrites 90.8%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 52000000:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\frac{\left(z + 1\right) - z}{\sqrt{z + 1} + \sqrt{z}} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + \sqrt{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right) + t\_1\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ z 1.0)) (sqrt z))))
   (if (<= t_1 2e-5)
     (+
      (- (fma (sqrt (/ 1.0 z)) 0.5 (sqrt (+ 1.0 y))) (+ (sqrt y) (sqrt x)))
      (sqrt (+ 1.0 x)))
     (+
      (+ (+ (- 1.0 (sqrt y)) (- 1.0 (sqrt x))) t_1)
      (- (sqrt (+ t 1.0)) (sqrt t))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double tmp;
	if (t_1 <= 2e-5) {
		tmp = (fma(sqrt((1.0 / z)), 0.5, sqrt((1.0 + y))) - (sqrt(y) + sqrt(x))) + sqrt((1.0 + x));
	} else {
		tmp = (((1.0 - sqrt(y)) + (1.0 - sqrt(x))) + t_1) + (sqrt((t + 1.0)) - sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	tmp = 0.0
	if (t_1 <= 2e-5)
		tmp = Float64(Float64(fma(sqrt(Float64(1.0 / z)), 0.5, sqrt(Float64(1.0 + y))) - Float64(sqrt(y) + sqrt(x))) + sqrt(Float64(1.0 + x)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(y)) + Float64(1.0 - sqrt(x))) + t_1) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-5], N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 2.00000000000000016e-5

   1. Initial program 87.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 4.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 19.7%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{y + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in z around inf

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 29.8%

       \[\leadsto \sqrt{x + 1} + \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   if 2.00000000000000016e-5 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-sqrt.f64 53.4

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 53.4%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right) + \color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right) + \color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1 - \sqrt{x}\right) + \color{blue}{\sqrt{\frac{1}{y}}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 28.3

         \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right) + \sqrt{\color{blue}{\frac{1}{y}}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 28.3%

      \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right) + \color{blue}{\sqrt{\frac{1}{y}} \cdot 0.5}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \sqrt{\frac{1}{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \sqrt{\frac{1}{y}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-sqrt.f64 27.0

          \[\leadsto \left(\left(\left(1 - \color{blue}{\sqrt{x}}\right) + \sqrt{\frac{1}{y}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Applied rewrites 27.0%

       \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \sqrt{\frac{1}{y}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. Taylor expanded in y around 0

       \[\leadsto \left(\left(\left(1 - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   13. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \left(\left(\left(1 - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-sqrt.f64 27.4

          \[\leadsto \left(\left(\left(1 - \sqrt{x}\right) + \left(1 - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   14. Applied rewrites 27.4%

       \[\leadsto \left(\left(\left(1 - \sqrt{x}\right) + \color{blue}{\left(1 - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + \sqrt{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{y}\right) + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{y} + \sqrt{x}\\ t_3 := \sqrt{z + 1}\\ t_4 := \sqrt{1 + x}\\ \mathbf{if}\;t\_3 - \sqrt{z} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, t\_1\right) - t\_2\right) + t\_4\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t\_3 + t\_4\right) + t\_1\right) - \sqrt{z}\right) - t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (+ (sqrt y) (sqrt x)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (sqrt (+ 1.0 x))))
   (if (<= (- t_3 (sqrt z)) 2e-5)
     (+ (- (fma (sqrt (/ 1.0 z)) 0.5 t_1) t_2) t_4)
     (- (- (+ (+ t_3 t_4) t_1) (sqrt z)) t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt(y) + sqrt(x);
	double t_3 = sqrt((z + 1.0));
	double t_4 = sqrt((1.0 + x));
	double tmp;
	if ((t_3 - sqrt(z)) <= 2e-5) {
		tmp = (fma(sqrt((1.0 / z)), 0.5, t_1) - t_2) + t_4;
	} else {
		tmp = (((t_3 + t_4) + t_1) - sqrt(z)) - t_2;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(sqrt(y) + sqrt(x))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_3 - sqrt(z)) <= 2e-5)
		tmp = Float64(Float64(fma(sqrt(Float64(1.0 / z)), 0.5, t_1) - t_2) + t_4);
	else
		tmp = Float64(Float64(Float64(Float64(t_3 + t_4) + t_1) - sqrt(z)) - t_2);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 2e-5], N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision] + t$95$4), $MachinePrecision], N[(N[(N[(N[(t$95$3 + t$95$4), $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 2.00000000000000016e-5

   1. Initial program 87.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 4.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 19.7%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{y + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in z around inf

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 29.8%

       \[\leadsto \sqrt{x + 1} + \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   if 2.00000000000000016e-5 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 18.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 18.1%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} + \sqrt{1 + z}\right) + \sqrt{y + 1}\right) - \sqrt{z}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + \sqrt{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt{z + 1} + \sqrt{1 + x}\right) + \sqrt{1 + y}\right) - \sqrt{z}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{y} + \sqrt{x}\\ t_3 := \sqrt{z + 1}\\ t_4 := \sqrt{1 + x}\\ \mathbf{if}\;t\_3 - \sqrt{z} \leq 0.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, t\_1\right) - t\_2\right) + t\_4\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t\_4 - \sqrt{z}\right) - t\_2\right) + t\_3\right) + t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y)))
        (t_2 (+ (sqrt y) (sqrt x)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (sqrt (+ 1.0 x))))
   (if (<= (- t_3 (sqrt z)) 0.0001)
     (+ (- (fma (sqrt (/ 1.0 z)) 0.5 t_1) t_2) t_4)
     (+ (+ (- (- t_4 (sqrt z)) t_2) t_3) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt(y) + sqrt(x);
	double t_3 = sqrt((z + 1.0));
	double t_4 = sqrt((1.0 + x));
	double tmp;
	if ((t_3 - sqrt(z)) <= 0.0001) {
		tmp = (fma(sqrt((1.0 / z)), 0.5, t_1) - t_2) + t_4;
	} else {
		tmp = (((t_4 - sqrt(z)) - t_2) + t_3) + t_1;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(sqrt(y) + sqrt(x))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_3 - sqrt(z)) <= 0.0001)
		tmp = Float64(Float64(fma(sqrt(Float64(1.0 / z)), 0.5, t_1) - t_2) + t_4);
	else
		tmp = Float64(Float64(Float64(Float64(t_4 - sqrt(z)) - t_2) + t_3) + t_1);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision] + t$95$4), $MachinePrecision], N[(N[(N[(N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 1.00000000000000005e-4

   1. Initial program 87.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 4.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 19.7%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{y + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in z around inf

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 29.8%

       \[\leadsto \sqrt{x + 1} + \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   if 1.00000000000000005e-4 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 18.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 32.2%

      \[\leadsto \sqrt{y + 1} + \color{blue}{\left(\sqrt{1 + z} + \left(\left(\sqrt{x + 1} - \sqrt{z}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 0.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + \sqrt{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt{1 + x} - \sqrt{z}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + \sqrt{z + 1}\right) + \sqrt{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{z + 1}\\ \mathbf{if}\;t\_2 - \sqrt{z} \leq 0.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, t\_1\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + \sqrt{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 + 1\right) + t\_2\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))) (t_2 (sqrt (+ z 1.0))))
   (if (<= (- t_2 (sqrt z)) 0.0001)
     (+
      (- (fma (sqrt (/ 1.0 z)) 0.5 t_1) (+ (sqrt y) (sqrt x)))
      (sqrt (+ 1.0 x)))
     (- (+ (+ t_1 1.0) t_2) (+ (+ (sqrt z) (sqrt y)) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((z + 1.0));
	double tmp;
	if ((t_2 - sqrt(z)) <= 0.0001) {
		tmp = (fma(sqrt((1.0 / z)), 0.5, t_1) - (sqrt(y) + sqrt(x))) + sqrt((1.0 + x));
	} else {
		tmp = ((t_1 + 1.0) + t_2) - ((sqrt(z) + sqrt(y)) + sqrt(x));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (Float64(t_2 - sqrt(z)) <= 0.0001)
		tmp = Float64(Float64(fma(sqrt(Float64(1.0 / z)), 0.5, t_1) - Float64(sqrt(y) + sqrt(x))) + sqrt(Float64(1.0 + x)));
	else
		tmp = Float64(Float64(Float64(t_1 + 1.0) + t_2) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 + 1.0), $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 1.00000000000000005e-4

   1. Initial program 87.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 4.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 19.7%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{y + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in z around inf

      \[\leadsto \sqrt{x + 1} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 29.8%

       \[\leadsto \sqrt{x + 1} + \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   if 1.00000000000000005e-4 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 18.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 16.5%

      \[\leadsto \left(\left(\sqrt{y + 1} + 1\right) + \sqrt{z + 1}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 0.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + \sqrt{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + y} + 1\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{y} + \sqrt{x}\\ t_3 := \sqrt{1 + x}\\ t_4 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;y \leq 5.9 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(2 + \mathsf{fma}\left(0.5, y, \frac{1}{t\_1 + \sqrt{z}}\right)\right) - t\_2\right) + t\_4\\ \mathbf{elif}\;y \leq 14500000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sqrt{1 + y} + t\_3\right) - t\_2, z, 0.5 \cdot \sqrt{z}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + t\_3} + \left(t\_1 - \sqrt{z}\right)\right) + t\_4\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0)))
        (t_2 (+ (sqrt y) (sqrt x)))
        (t_3 (sqrt (+ 1.0 x)))
        (t_4 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= y 5.9e-8)
     (+ (- (+ 2.0 (fma 0.5 y (/ 1.0 (+ t_1 (sqrt z))))) t_2) t_4)
     (if (<= y 14500000000000.0)
       (/ (fma (- (+ (sqrt (+ 1.0 y)) t_3) t_2) z (* 0.5 (sqrt z))) z)
       (+ (+ (/ 1.0 (+ (sqrt x) t_3)) (- t_1 (sqrt z))) t_4)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = sqrt(y) + sqrt(x);
	double t_3 = sqrt((1.0 + x));
	double t_4 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if (y <= 5.9e-8) {
		tmp = ((2.0 + fma(0.5, y, (1.0 / (t_1 + sqrt(z))))) - t_2) + t_4;
	} else if (y <= 14500000000000.0) {
		tmp = fma(((sqrt((1.0 + y)) + t_3) - t_2), z, (0.5 * sqrt(z))) / z;
	} else {
		tmp = ((1.0 / (sqrt(x) + t_3)) + (t_1 - sqrt(z))) + t_4;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = Float64(sqrt(y) + sqrt(x))
	t_3 = sqrt(Float64(1.0 + x))
	t_4 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (y <= 5.9e-8)
		tmp = Float64(Float64(Float64(2.0 + fma(0.5, y, Float64(1.0 / Float64(t_1 + sqrt(z))))) - t_2) + t_4);
	elseif (y <= 14500000000000.0)
		tmp = Float64(fma(Float64(Float64(sqrt(Float64(1.0 + y)) + t_3) - t_2), z, Float64(0.5 * sqrt(z))) / z);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(x) + t_3)) + Float64(t_1 - sqrt(z))) + t_4);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.9e-8], N[(N[(N[(2.0 + N[(0.5 * y + N[(1.0 / N[(t$95$1 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[y, 14500000000000.0], N[(N[(N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + t$95$3), $MachinePrecision] - t$95$2), $MachinePrecision] * z + N[(0.5 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.8999999999999999e-8

   1. Initial program 97.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 98.1

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 98.1%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot y + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot y + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 58.1%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{x + 1} + 1\right) + \mathsf{fma}\left(0.5, y, \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(\left(2 + \left(\frac{1}{2} \cdot y + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 49.6%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.5, y, \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right) + 2\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 5.8999999999999999e-8 < y < 1.45e13

   1. Initial program 81.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 17.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 19.5%

      \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto \frac{\frac{1}{2} \cdot \sqrt{z} + z \cdot \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)}{z} \]
   10. Step-by-step derivation

   11. Applied rewrites 21.2%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{y + 1} + \sqrt{x + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right), z, 0.5 \cdot \sqrt{z}\right)}{z} \]

   if 1.45e13 < y

   1. Initial program 87.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. flip-- N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 88.9%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(y + 1\right) - y, \sqrt{x} + \sqrt{1 + x}, \left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\left(1 + x\right) - x\right)\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\color{blue}{1 + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{1 + y}\right)\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right)} + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\color{blue}{\sqrt{y}} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \color{blue}{\sqrt{1 + y}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{y + 1}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 90.5

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{y + 1}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 90.5%

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{y + 1}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 89.7

         \[\leadsto \left(\frac{1}{\sqrt{x + 1} + \color{blue}{\sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied rewrites 89.7%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.9 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(2 + \mathsf{fma}\left(0.5, y, \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;y \leq 14500000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right), z, 0.5 \cdot \sqrt{z}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{z + 1}\\ \mathbf{if}\;t\_2 - \sqrt{z} \leq 0.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, t\_1\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 + 1\right) + t\_2\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))) (t_2 (sqrt (+ z 1.0))))
   (if (<= (- t_2 (sqrt z)) 0.0001)
     (+ (- (fma (sqrt (/ 1.0 z)) 0.5 t_1) (+ (sqrt y) (sqrt x))) 1.0)
     (- (+ (+ t_1 1.0) t_2) (+ (+ (sqrt z) (sqrt y)) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((z + 1.0));
	double tmp;
	if ((t_2 - sqrt(z)) <= 0.0001) {
		tmp = (fma(sqrt((1.0 / z)), 0.5, t_1) - (sqrt(y) + sqrt(x))) + 1.0;
	} else {
		tmp = ((t_1 + 1.0) + t_2) - ((sqrt(z) + sqrt(y)) + sqrt(x));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (Float64(t_2 - sqrt(z)) <= 0.0001)
		tmp = Float64(Float64(fma(sqrt(Float64(1.0 / z)), 0.5, t_1) - Float64(sqrt(y) + sqrt(x))) + 1.0);
	else
		tmp = Float64(Float64(Float64(t_1 + 1.0) + t_2) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(t$95$1 + 1.0), $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 1.00000000000000005e-4

   1. Initial program 87.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 4.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 19.8%

      \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 30.8%

       \[\leadsto 1 + \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   if 1.00000000000000005e-4 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 18.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 16.5%

      \[\leadsto \left(\left(\sqrt{y + 1} + 1\right) + \sqrt{z + 1}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 0.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{1 + y} + 1\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{z + 1}\\ \mathbf{if}\;t\_2 - \sqrt{z} \leq 0.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, t\_1\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_2 + t\_1\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))) (t_2 (sqrt (+ z 1.0))))
   (if (<= (- t_2 (sqrt z)) 0.0001)
     (+ (- (fma (sqrt (/ 1.0 z)) 0.5 t_1) (+ (sqrt y) (sqrt x))) 1.0)
     (+ (- (+ t_2 t_1) (+ (+ (sqrt z) (sqrt y)) (sqrt x))) 1.0))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((z + 1.0));
	double tmp;
	if ((t_2 - sqrt(z)) <= 0.0001) {
		tmp = (fma(sqrt((1.0 / z)), 0.5, t_1) - (sqrt(y) + sqrt(x))) + 1.0;
	} else {
		tmp = ((t_2 + t_1) - ((sqrt(z) + sqrt(y)) + sqrt(x))) + 1.0;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (Float64(t_2 - sqrt(z)) <= 0.0001)
		tmp = Float64(Float64(fma(sqrt(Float64(1.0 / z)), 0.5, t_1) - Float64(sqrt(y) + sqrt(x))) + 1.0);
	else
		tmp = Float64(Float64(Float64(t_2 + t_1) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x))) + 1.0);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(t$95$2 + t$95$1), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 1.00000000000000005e-4

   1. Initial program 87.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 4.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 19.8%

      \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 30.8%

       \[\leadsto 1 + \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   if 1.00000000000000005e-4 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 18.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 22.1%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{y + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{y + 1}\right)} - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 21.6%

       \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{y + 1}\right)} - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 0.0001:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{z + 1} + \sqrt{1 + y}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{y} + \sqrt{x}\\ \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, t\_1\right) - t\_2\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + \sqrt{1 + x}\right) - t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 y))) (t_2 (+ (sqrt y) (sqrt x))))
   (if (<= (- (sqrt (+ z 1.0)) (sqrt z)) 0.02)
     (+ (- (fma (sqrt (/ 1.0 z)) 0.5 t_1) t_2) 1.0)
     (- (+ t_1 (sqrt (+ 1.0 x))) t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt(y) + sqrt(x);
	double tmp;
	if ((sqrt((z + 1.0)) - sqrt(z)) <= 0.02) {
		tmp = (fma(sqrt((1.0 / z)), 0.5, t_1) - t_2) + 1.0;
	} else {
		tmp = (t_1 + sqrt((1.0 + x))) - t_2;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = Float64(sqrt(y) + sqrt(x))
	tmp = 0.0
	if (Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) <= 0.02)
		tmp = Float64(Float64(fma(sqrt(Float64(1.0 / z)), 0.5, t_1) - t_2) + 1.0);
	else
		tmp = Float64(Float64(t_1 + sqrt(Float64(1.0 + x))) - t_2);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(t$95$1 + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 0.0200000000000000004

   1. Initial program 87.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 4.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 19.4%

      \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 30.3%

       \[\leadsto 1 + \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

   if 0.0200000000000000004 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 97.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 18.4%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 8.3%

      \[\leadsto \left(\sqrt{y + 1} + \sqrt{x + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 66.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{x}\right) + \sqrt{\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 4e+23)
   (- (+ (sqrt (+ 1.0 y)) (sqrt (+ 1.0 x))) (+ (sqrt y) (sqrt x)))
   (+ (- (sqrt x)) (sqrt (fma (sqrt x) (sqrt x) 1.0)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4e+23) {
		tmp = (sqrt((1.0 + y)) + sqrt((1.0 + x))) - (sqrt(y) + sqrt(x));
	} else {
		tmp = -sqrt(x) + sqrt(fma(sqrt(x), sqrt(x), 1.0));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4e+23)
		tmp = Float64(Float64(sqrt(Float64(1.0 + y)) + sqrt(Float64(1.0 + x))) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(-sqrt(x)) + sqrt(fma(sqrt(x), sqrt(x), 1.0)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 4e+23], N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[x], $MachinePrecision]) + N[Sqrt[N[(N[Sqrt[x], $MachinePrecision] * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.9999999999999997e23

   1. Initial program 96.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 18.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 21.5%

      \[\leadsto \left(\sqrt{y + 1} + \sqrt{x + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

   if 3.9999999999999997e23 < y

   1. Initial program 87.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 3.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 19.2%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{y + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \sqrt{x + 1} + -1 \cdot \color{blue}{\sqrt{x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.9%

       \[\leadsto \sqrt{x + 1} + \left(-\sqrt{x}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 18.9%

       \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, 1\right)} + \left(-\sqrt{\color{blue}{x}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\left(\sqrt{1 + y} + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{x}\right) + \sqrt{\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 64.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right) + \sqrt{1 + x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ (- (sqrt (+ 1.0 y)) (+ (sqrt y) (sqrt x))) (sqrt (+ 1.0 x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (sqrt((1.0 + y)) - (sqrt(y) + sqrt(x))) + sqrt((1.0 + x));
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (sqrt((1.0d0 + y)) - (sqrt(y) + sqrt(x))) + sqrt((1.0d0 + x))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return (Math.sqrt((1.0 + y)) - (Math.sqrt(y) + Math.sqrt(x))) + Math.sqrt((1.0 + x));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (math.sqrt((1.0 + y)) - (math.sqrt(y) + math.sqrt(x))) + math.sqrt((1.0 + x))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(sqrt(Float64(1.0 + y)) - Float64(sqrt(y) + sqrt(x))) + sqrt(Float64(1.0 + x)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (sqrt((1.0 + y)) - (sqrt(y) + sqrt(x))) + sqrt((1.0 + x));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.4%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
4. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   14. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   16. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

5. Applied rewrites 11.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 20.9%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{y + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

8. Taylor expanded in z around inf

   \[\leadsto \sqrt{x + 1} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]
9. Step-by-step derivation

10. Applied rewrites 20.3%

    \[\leadsto \sqrt{x + 1} + \left(\sqrt{y + 1} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

11. Final simplification 20.3%

    \[\leadsto \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right) + \sqrt{1 + x} \]
12. Add Preprocessing

Alternative 18: 36.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(-\sqrt{x}\right) + \sqrt{\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, 1\right)} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ (- (sqrt x)) (sqrt (fma (sqrt x) (sqrt x) 1.0))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return -sqrt(x) + sqrt(fma(sqrt(x), sqrt(x), 1.0));
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(-sqrt(x)) + sqrt(fma(sqrt(x), sqrt(x), 1.0)))
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[((-N[Sqrt[x], $MachinePrecision]) + N[Sqrt[N[(N[Sqrt[x], $MachinePrecision] * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.4%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
4. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   14. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   16. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

5. Applied rewrites 11.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 20.9%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{y + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

8. Taylor expanded in x around inf

   \[\leadsto \sqrt{x + 1} + -1 \cdot \color{blue}{\sqrt{x}} \]
9. Step-by-step derivation

10. Applied rewrites 14.7%

    \[\leadsto \sqrt{x + 1} + \left(-\sqrt{x}\right) \]
11. Step-by-step derivation

12. Applied rewrites 14.7%

    \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, 1\right)} + \left(-\sqrt{\color{blue}{x}}\right) \]

13. Final simplification 14.7%

    \[\leadsto \left(-\sqrt{x}\right) + \sqrt{\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, 1\right)} \]
14. Add Preprocessing

Alternative 19: 36.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(-\sqrt{x}\right) + \sqrt{1 + x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (+ (- (sqrt x)) (sqrt (+ 1.0 x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return -sqrt(x) + sqrt((1.0 + x));
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -sqrt(x) + sqrt((1.0d0 + x))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return -Math.sqrt(x) + Math.sqrt((1.0 + x));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return -math.sqrt(x) + math.sqrt((1.0 + x))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(-sqrt(x)) + sqrt(Float64(1.0 + x)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = -sqrt(x) + sqrt((1.0 + x));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[((-N[Sqrt[x], $MachinePrecision]) + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.4%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
4. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   14. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   16. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

5. Applied rewrites 11.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 20.9%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{y + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

8. Taylor expanded in x around inf

   \[\leadsto \sqrt{x + 1} + -1 \cdot \color{blue}{\sqrt{x}} \]
9. Step-by-step derivation

10. Applied rewrites 14.7%

    \[\leadsto \sqrt{x + 1} + \left(-\sqrt{x}\right) \]

11. Final simplification 14.7%

    \[\leadsto \left(-\sqrt{x}\right) + \sqrt{1 + x} \]
12. Add Preprocessing

Alternative 20: 35.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \mathsf{fma}\left(0.5, x, 1\right) + \left(-\sqrt{x}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (+ (fma 0.5 x 1.0) (- (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return fma(0.5, x, 1.0) + -sqrt(x);
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(fma(0.5, x, 1.0) + Float64(-sqrt(x)))
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(0.5 * x + 1.0), $MachinePrecision] + (-N[Sqrt[x], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 92.4%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
4. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   14. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   16. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

5. Applied rewrites 11.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 20.9%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{y + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

8. Taylor expanded in x around inf

   \[\leadsto \sqrt{x + 1} + -1 \cdot \color{blue}{\sqrt{x}} \]
9. Step-by-step derivation

10. Applied rewrites 14.7%

    \[\leadsto \sqrt{x + 1} + \left(-\sqrt{x}\right) \]

11. Taylor expanded in x around 0

    \[\leadsto \left(1 + \frac{1}{2} \cdot x\right) + \left(-\color{blue}{\sqrt{x}}\right) \]
12. Step-by-step derivation

13. Applied rewrites 15.1%

    \[\leadsto \mathsf{fma}\left(0.5, x, 1\right) + \left(-\color{blue}{\sqrt{x}}\right) \]
14. Add Preprocessing

Alternative 21: 35.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(-\sqrt{x}\right) + 1 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (+ (- (sqrt x)) 1.0))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return -sqrt(x) + 1.0;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -sqrt(x) + 1.0d0
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return -Math.sqrt(x) + 1.0;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return -math.sqrt(x) + 1.0

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(-sqrt(x)) + 1.0)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = -sqrt(x) + 1.0;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[((-N[Sqrt[x], $MachinePrecision]) + 1.0), $MachinePrecision]

Derivation

1. Initial program 92.4%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
4. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   14. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   16. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

5. Applied rewrites 11.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 20.9%

   \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\sqrt{1 + z} + \sqrt{y + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} \]

8. Taylor expanded in x around inf

   \[\leadsto \sqrt{x + 1} + -1 \cdot \color{blue}{\sqrt{x}} \]
9. Step-by-step derivation

10. Applied rewrites 14.7%

    \[\leadsto \sqrt{x + 1} + \left(-\sqrt{x}\right) \]

11. Taylor expanded in x around 0

    \[\leadsto 1 + \left(-\color{blue}{\sqrt{x}}\right) \]
12. Step-by-step derivation

13. Applied rewrites 13.3%

    \[\leadsto 1 + \left(-\color{blue}{\sqrt{x}}\right) \]

14. Final simplification 13.3%

    \[\leadsto \left(-\sqrt{x}\right) + 1 \]
15. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+
    (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
    (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + (1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y)))) + (1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z)))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + (1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y)))) + (1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z)))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + (1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y)))) + (1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z)))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024268 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (/ 1 (+ (sqrt (+ x 1)) (sqrt x))) (/ 1 (+ (sqrt (+ y 1)) (sqrt y)))) (/ 1 (+ (sqrt (+ z 1)) (sqrt z)))) (- (sqrt (+ t 1)) (sqrt t))))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))